List of Tables
3.2 Modeling of the travel behavior
Discrete choice models describe, explain, and predict the choices between two or more alternatives, such as choosing between the modes of transport or choosing from various brands of goods. Discrete choice models statistically relate the choice made by each person to the attributes of the person and the attributes of the alternatives available to the person.
In the standard logit modeling framework, the probability of an individual choosing an alternative is given as below;
( )
∑ (3.1) where, Pr(i) is the probability of decision maker choosing an alternative i; Vj is the systematic component of the utility of alternative j. MNL model has three main drawbacks, namely, inability to explain the unobserved component of the taste variation, unrealistic proportional substitution across alternatives, and the inability to explain the correlation of the unobserved factors which may result in case of panel data (Train, 2009).
Sample collected in the present study is not stratified based on the socioeconomic characteristics. Given the wide variations in the socioeconomic characteristics of the sampled individuals, it is expected to have unobserved taste heterogeneity among the sampled individuals. Since the choice set contains many similar alternatives, proportional substitution pattern may also not hold well. Also, in the present study, the choice model has been estimated with both the SP and RP data. In case of SP data, there can be significant correlations among the unobserved components of the modal utilities corresponding to various choice scenarios.
These drawbacks can be handled by either GEV class models such as the nested logit model or more flexible mixed logit models. Mixed logit model generalises a standard logit model by allowing a parameter (i.e., coefficient) associated with each observed attribute to vary randomly across the individuals. In case of random parameter logit model, the utility derived by person n from alternative j is specified as;
Unj = (3.2)
where, are the observable attributes of the alternative and decision maker;
is a vector of coefficients of variables for person n representating the person’s taste; is the random term that is independently and identically distributed (IID) extreme value. Here the coefficients vary across the decision makers in the population with density f(β). The specification is same as for standard logit model except that β varies over decision makers rather than being fixed. The probability conditional on β is given by;
Lni ( ) =
∑ (3.3)
where, Lni ( ) is the logit probability evaluated at parameter .
As the researcher does not observe and, therefore, cannot condition on . The unconditional choice probability over all values of is expressed as
Pni = ( )
∑ ) ( ) (3.4) where, θ refers collectively to the parameters of chosen distribution (such as mean and variance of β).
A mixed logit model can also be used without a random-cofficient interpretation, as simply representing the error components that create correlations among the unobserved utilities for different alternatives (Train, 2009). The utility in this case is specified as;
Unj = (3.5) where,
Xnj and Znj are the vectors of observable variables related to alternative j, is a vector of fixed coefficients,
is a vector of random terms with zero mean, and,
εnj is error component which is iid extreme value.
The terms in Znj are error components that, along with εnj forms the stochastic portion of utility. The unobserved component of the utility is , which can be correlated over alternatives depending upon the specification of Znj. In case of standard logit model Znj is zero and there is no correlation in the utility over the alternatives.
The unconditional choice probability as given in equation 3.7 cannot be calculated exactly as the integral is not in closed form. Probabilities are approximated through simulation for a given value of θ. For simulation, a value of β from ( ) is drawn and logit formula Lni ( ) is calculated. These steps are repeated and the results are averaged.
This average is the simulated probability and is given as (Train, 2009);
̂ ∑ ( ) (3.6) where, R is the number of draws. ̂ is the unbaised estimator of Pni . Its variance decreases as R increases and is positive so that ln ̂ is defined.The simulated probalities are inserted into log likelihood function to get the simulated log likelihood.
3.2.1 Handling of SP data in mixed logit framework
Data from the SP survey contains repeated choice data from a single individual. The number of choice situations can vary over people, and choice sets can vary over people and choice situations. Mixed logit model with panel specification can handle the correlations among the unobserved error components, mainly associated with the SP data. In case of repeated choice data, as in case of panel data, utility from alternative j in choice situation t by an individual n is given as (Revelt and Train, 1998);
(3.7) where , is iid extreme value over individual, alternative, and time.
Conditional on β, the probability that the person making the sequence of choices is the product of logit formulas given below;
( ) ∏ [
∑ ]
(3.8)
The unconditional probability is the integral of this product over all values of .
( ) ( ) . (3.9) The simulated probabilities and the simulated likelihood functions can be obtained following the procedure discussed in the previous sub-section.
3.2.2 Handling of combined SP-RP data
When two sources of data, SP and RP, are used to estimate the choice models there will be issues with the scales of the data. Combining two data sources involves imposing the restriction that the common attributes (atleast one) have the same parameters in both the data sources, i.e., βRP = βSP = β. Data enrichment includes pooling of two choice data sources under the restriction that common parameters are equal while controlling for the scale factors. Scales of the utilities, corresponding to the SP and the RP data would be different as the unobserved variance may not be same in the utilities corresponding to these data. Since it is difficult to identify both the scale parameters, it is conventional to fix the scale of RP data set (λRP = 1) and estimate the λSP, which represents the relative scale. The corresponding choice model may be written as follows (Louviere et al. 2000);
( )
∑ ( ) (3.10) [ ( )]
∑ [ ( )] (3.11) where, i is an alternative in choice sets CSP or CRP, α’s are the data source-specific alternative specific constants (ASCs)and βRP and βSP are the utility parameters for attributes common to both the data sets, and δ and ω are utility parameters for unique attributes in each data set. and are the scale factors.
So the deterministic utility portion for a mode for both the RP and SP parts can be written as
( ) (3.12)